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Mirrors > Home > MPE Home > Th. List > ficardun2 | Structured version Visualization version GIF version |
Description: The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.) |
Ref | Expression |
---|---|
ficardun2 | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undjudom 9581 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | |
2 | finnum 9365 | . . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
3 | finnum 9365 | . . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
4 | cardadju 9608 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) | |
5 | 2, 3, 4 | syl2an 595 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) |
6 | domentr 8556 | . . . 4 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))) | |
7 | 1, 5, 6 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))) |
8 | unfi 8773 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
9 | finnum 9365 | . . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∪ 𝐵) ∈ dom card) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ dom card) |
11 | ficardom 9378 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
12 | ficardom 9378 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω) | |
13 | nnacl 8226 | . . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω) | |
14 | 11, 12, 13 | syl2an 595 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω) |
15 | nnon 7575 | . . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On) | |
16 | onenon 9366 | . . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ On → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) | |
17 | 14, 15, 16 | 3syl 18 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) |
18 | carddom2 9394 | . . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))) | |
19 | 10, 17, 18 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))) |
20 | 7, 19 | mpbird 258 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵)))) |
21 | cardnn 9380 | . . 3 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵))) | |
22 | 14, 21 | syl 17 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵))) |
23 | 20, 22 | sseqtrd 4004 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 ωcom 7569 +o coa 8088 ≈ cen 8494 ≼ cdom 8495 Fincfn 8497 ⊔ cdju 9315 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 |
This theorem is referenced by: (None) |
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